# Linear algebra

How to construct a matrix that has column space with vectors: 1 1 0 and 0 1 1 and nullspace with vectors 1 0 1 and 0 0 1?
(edited 4 months ago)
Original post by Student765432101
How to construct a matrix that has column space with vectors: 1 1 0 and 0 1 1 and nullspace with vectors 1 0 1 and 0 0 1?

Any ideas, what do you understand about the terms? It doesnt sound possible to me.
(edited 4 months ago)
Original post by mqb2766
Any ideas, what do you understand about the terms? It doesnt sound possible to me.

Well, the professor gave us an example when the null space has 1 vector, the example goes like this: Construct a matrix whose column space contains vectors (1, 1, 5) and (0, 3, 1), and whose null space contains the vector (1, 1, 2). This one is easy, obviously the matrix will have 3 rows and 3 columns, the third column has unknown elements a13 a23 and a33. We multiply the matrix with the vector (1,1,2) to get (0,0,0) and the solution is easy. a13= -1/2, a23=-2 and a33=-3. But I don't know how to do it when the null space has 2 vectors.
Original post by Student765432101
Well, the professor gave us an example when the null space has 1 vector, the example goes like this: Construct a matrix whose column space contains vectors (1, 1, 5) and (0, 3, 1), and whose null space contains the vector (1, 1, 2). This one is easy, obviously the matrix will have 3 rows and 3 columns, the third column has unknown elements a13 a23 and a33. We multiply the matrix with the vector (1,1,2) to get (0,0,0) and the solution is easy. a13= -1/2, a23=-2 and a33=-3. But I don't know how to do it when the null space has 2 vectors.

Agree with the example you were given and your uncertainty about the question. Youve a 3*3 matrix again, so construct the first two columns (column space) then argue about the third one (nullspace), or argue that its not possible.

If youve come across the rank nullity theorem, the argument is straightforward, but its not much more work to try and construct the third column.
(edited 4 months ago)
Original post by mqb2766
Agree with the example you were given and your uncertainty about the question. Youve a 3*3 matrix again, so construct the first two columns (column space) then argue about the third one (nullspace).

If youve come across the rank nullity theorem, the argument is straightforward, but its not much more work to try and construct the third column.

I dont understand, how can I multiply the 3x3 matrix (column space) by 3x2 matrix (the nullspace) to get zero vector... it doesn't seem possible.
Original post by Student765432101
I dont understand, how can I multiply the 3x3 matrix (column space) by 3x2 matrix (the nullspace) to get zero vector... it doesn't seem possible.

It seemed possible when the nullspace was only 1 vector... but how do I do this one?
Original post by Student765432101
I dont understand, how can I multiply the 3x3 matrix (column space) by 3x2 matrix (the nullspace) to get zero vector... it doesn't seem possible.

I agree, its not possible. If you try and formulate the equations for the third column values as per your reply, theyd be inconsistent .
Original post by mqb2766
I agree, its not possible. If you try and formulate the equations for the third column values as per your reply, theyd be inconsistent .

Okay, thank you for your time.